Recover the whole from a fractional part

3.NF.A.13.OA.A.4 · take · grade 3

Archetype: Part-Whole Fraction Reasoning · step in a 5-type progression

▶ Practice — 12 problems

Find $\frac{1}{9}$ of the number $\star$ that satisfies the following.

$\frac{7}{12} \text{ of } \star \text{ is } 21.$

Show solution

Understand

A mystery number star has the property that 7/12 of it equals 21. I first need to find star, then report 1/9 of star.

Givens

7/12 of star is 21.
star is the whole quantity (the 'whole') being split into twelfths.

Unknowns

The value of 1/9 of star.

Constraints

7/12 of star means star split into 12 equal parts, with 7 of those parts totaling 21.
star must come out a whole number so 1/9 of it makes sense.

Plan

#11 Work Backwards · also uses: #7 Identify Subproblems

I work backwards from the part to the whole: if 7 twelfths equal 21, then one twelfth equals 21/7, and the whole is 12 of those. Then taking 1/9 of the recovered whole is a second small subproblem.

Execute

#11 Work Backwards 3.OA.A.4

7 equal twelfths make 21, so one twelfth is 21 divided by 7.

21 \div 7 = 3

If 7 equal parts total 21, each part is found by sharing 21 into 7 equal pieces.

#11 Work Backwards 3.NF.A.1

star is made of 12 such twelfths, so multiply one twelfth (3) by 12.

3 \times 12 = 36

The whole is just all 12 of its equal parts put back together.

#7 Identify Subproblems 3.NF.A.1

Now find 1/9 of 36 by dividing 36 into 9 equal parts.

36 \div 9 = 4

A unit fraction 1/9 of a number is that number shared into 9 equal parts.

Answer: 4

Review

Check: 7/12 of 36 is 36 ÷ 12 × 7 = 3 × 7 = 21, which matches the given. And 1/9 of 36 is 4. Both come out as clean whole numbers, so 36 is the right whole and 4 is the answer.

Convert to algebra (tool 13): (7/12) x star = 21 gives star = 21 x 12 / 7 = 36, then star / 9 = 4 — same result, but the work-backwards unit-fraction route avoids algebra.

Standards · min grade 3

3.OA.A.4 Determine unknown whole number in multiplication or division equation — Finding one twelfth from 7 twelfths = 21 (an unknown-part division).
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Rebuilding the whole from twelfths and taking 1/9 of it.

💡 This only needs Grade 3 fraction sense: find one equal part, build the whole, then split again!